Equitable distribution of resources in Transportation Networks · Optimization model: Convex relaxation We replace variables V with W variables and to make the substitution equivalent

Equitable distribution of resources in Transportation Networks

ĽUBOŠ BUZNAUNIVERSITY OF ŽILINA

[email protected]://frdsa.uniza.sk/~buzna

WASACNA 2016: Tatranské Matliare, 17. Sept. 2016

mailto:[email protected]://frdsa.uniza.sk/~buzna

University of Žilina, Faculty of Management Science and InformaticsDepartment of Mathematical Methods and Operations Research

Head: Prof. Ing. Ľudmila Jánošíková, PhD.

Web: http://frdsa.fri.uniza.sk/wordpress/

Education: Math courses for engineers (Algebra, Mathematical Analysis, Graph Theory, Game theory), Optimisation Methods, Data structures, Computer simulations, Geographical Information Systems, Cryptography,

Research: Optimisations and simulations of complex technological systems (private and public service systems, transportation networks, production systems, crisis management)

University of Žilina, University Science parkERA chair in Intelligent Transport Systems

Chair holder: Dr. Karl Ernst Ambrosh

FP 7 Project: ERAdiate Web: http://www.erachair.uniza.sk/

Research fields:

1. Cooperative ITS (eCall, localisation, standardisation)

2. Decarbonisation of Mobility (eMobility, GHG indicators, modal shift, simulation, mathematical modelling)

3. Urban Mobility / Smart City (urban mobility indicators, sensor networks, incident detection, urban logistics)

4. Intermodal ITS (modal shift, big data, MaaS, SUMP, modelling and simulation)

This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no.621386

Equitable distribution of resources in Transportation Networks

1. Theoretical background

2. Application I: Resilience of Natural Gas Networks During Conflicts, Crises and Disruptions

3. Application II: Design of Public Service Systems

4. Application III: Coordination of EVs Charging in the Distribution Networks

5. Current and future work


We consider a set of users . User assigns to a decision vector a utility .

User utilities form the performance vector:

Let for and in vector notation for Let the feasible outcome vectors be .Terminology: - is weakly preferred over - is strictly preferred over - and are equally preferred[H.Luss, Equitable Resource Allocation: Models, Algorithms and Applications, John Wiley & Sons, 2012]

J jx U j(x )

U (x )=(U 1( x) ,U 2(x ), ... ,U n(x ))

u j=U j (x) j=1, ..., n u=f ( x)

u1≼u2 u1 u2

u1≺u2 u1 u2

u1≃u2 u1 u2

U


Equitable solutions (fairness schemes) should (ideally) satisfy the following properties (axioms):

1. Completeness: either or for any 2. Transitivity: If and , then for any 3. Strictly monotonic: for any and ,where is the jth unit vector and is an arbitrary small positive constant.4. Scale invariance: If , then for any and .5. Anonymity (Impartiality): if is the permutation of the elements of for any[H.Luss, Equitable Resource Allocation: Models, Algorithms and Applications, John Wiley & Sons, 2012]

u1≼u2 u2≼u1 u1 ,u2∈Uu1≼u2 u2≼u3 u1≼u3 u1 ,u2 ,u3∈U

u−ϵ e j≺u (u−ϵ e j) ,u∈Uj=1, ..., n e j ϵ

u1≼u2 c u1≼c u2 u1 ,u2∈Uc>0

u1≃u2

u1 ,u2∈Uu1

u2


Equitable solutions (fairness schemes) should (ideally) satisfy the following properties (axioms):

6. Principle of Transferability: If , then for any and .

[H.Luss, Equitable Resource Allocation: Models, Algorithms and Applications, John Wiley & Sons, 2012]

u j1>u j2 u−ϵ e j 1+ϵ e j2≺u0


Fairness can be captured by combining a family of user utility functions, called -fair:

with optimisation: maximise

Subject to - utilitarian solution

(system optimum, max-flow) - proportional fairness - max-min fairness

[H.Luss, Equitable Resource Allocation: Models, Algorithms and Applications, John Wiley & Sons, 2012]

α

U j(x j ,α)={ x j1−α

1−αforα≥0,α≠0

log(x j)α=1

∑j=1

n

U j( x ,α)

x∈XFairness spectrum

FairnessEfficiency

α=0

α=1α→∞


Utilitarian solution (system optimum, max-flow):

-

The utilitarian solution maximises aggregate utility and hence it is important benchmark .

The utilitarian solution can leave some users with zero allocation and that can lead to large inequality. This can be considered unfair from the user point of view.

Property of the utilitarian solution: To increase an allocation by a we have to decrease a set of other allocations, such that the sum of of decreases is larger or equal to .

α=0



log(x j)α=1


Proportional fairness:

-

The proportionally fair allocation satisfies:

To increase the allocation by a percentage we have to decrease a set of other allocations, such that the sum of of percentage decreases is larger or equal to .

In communication networks, proportional fairness has emerged as a compromise between efficiency and fairness with an attractive interpretation in terms of shadow prices and market clearing mechanism.

[Frank P Kelly, Aman K Maulloo, and David KH Tan. “Rate control for communication networks: shadow prices, proportional fairness and stability”. In: Journal of the Operational Research society 49 (3) (1998), pp. 237–252.]

α=1



log(x j)α=1

∑j=1

n x j−x jPF

x jPF ≤0x j

PF


Max-min fairness:

-● Intuitively, max-min fairness recursively maximises the minimum

utility.● Allocation is max-min fair if no utility can be improved without

simultaneously decreasing another utility that is already smaller than or equal to the former (wealthy can only get wealthier by making poor even poorer).

● The computations involve a sequential optimisation procedure that identifies the corresponding utility levels at each step.

● Generalisation of the approach to discrete problems: Lexicographic minimax. From the vector we generate the vector by sorting performance functions in nonincreasing order. Then, we aim to find the solution corresponding to the lexicographically minimal vector.



log(x j)α=1α→∞

U (x )=(U 1( x) ,U 2(x ), ... ,U n(x ))U (n)(x )=(U j1(x ) ,U j2( x) , ...,U jn(x ))

Application I: Resilience of Natural Gas Networks During Conflicts, Crises and Disruptions

(joint work with Rui Carvalho, University of Durham, Flavio Bono, Marcelo Masera, Joint Research Center of EC, D. K. Arrowsmith, QMUL, Dirk Helbing, ETH Zurich

R. Carvalho, L. Buzna, F. Bono, M. Masera, D. K. Arrowsmith, D.Helbing, Resielience of Natural Gas Networks during Conflicts, Crises, and Disruptions, PLoS ONE 9(3): e90265, 2014

EXPOSURE

RESILIENCE

• Historical dependency of gas supply from few large sources leaves the European continent exposed to both a pipeline network that was not designed to transport large quantities of gas imported via Liquefied Natural Gas (LNG) terminals,

• and to the effects of political and social instabilities in countries that are heavily dependent either on the export of natural gas (eg Algeria, Libya, Qatar or Russia) or its transit (eg. Ukraine).

• Hence, it is challenging to build infrastructure that will be resilient to a wide range of possible crisis scenarios.

CRITICALITIES

Methodology

ModelModel

DataData

RoutingRoutingCongestion ControlCongestion Control

DataData

GIS data Gas trade network

Congestion Control

Congestion Control

Results: Global network throughput by scenario

Results: Resilience at country and network levels

• A country is resilient to crises if it combines high throughput per capita across scenarios with a low coefficient of variation of throughput.

• The network is resilient to a scenario if the vectors of country throughput per capita for the scenario and the baseline scenario are similar.


We analyse the country throughput per capita in the scenario space (20 scenarios) and in the country space (32 countries). The country groups highlighted in grey reflect a similar level of throughput per capita across scenarios;Countries belong to the high throughput per capita groups (dark grey) due to a combination of factors: diversity of supply; good access to network capacity (strategic geographical location); relatively small population.


• Larger values of the coefficient of variation indicate that country throughput varies across scenarios;

• Eastern European countries are sensitive to the scenarios where we hypothetically remove Russia or Ukraine –they are dependent on these countries;


Largest disruption

Most benignscenario

The present and future scenarios are clustered together when either Russia, Ukraine, the Netherlands, or Belarus are removed from the network;

Application II: Design of Public Service Systems

3 1

2

4

7

10

9

89

15

Sorted distances:1: 10,9,9,02: 15,9,7,0 3: 10,8,7,0 4: 15,9,8,0

L. Buzna, M. Kohani, J. Janacek, An Approximation Algorithm for the Facility Location Problem with Lexicographic Minimax Ordering, Journal of Applied Mathematics, Volume 2014 (2014), Article ID 562373, 12 pages

Set of feasible solutions

Algorithm A-LEX

Numerical experiments

● The algorithm found the optimal solution for all instances, where we could compare the results with the exact algorithm and it allows for solving larger problems● The algorithm A-LEX computed all small instances in the time which accounts for 42.5% and all medium instances for 47.6% of the time needed by the algorithm O-LEX ● Proposed approximation method is applicable to other types of similar problems with lexicographical minimax objective (e.g. maximum generalised assignment problem)

Application III: Coordination of EVs Charging in the Distribution Networks

(joint work with Rui Carvalho, University of Durham, Richard Gibbens and Frank Kelly, University of Cambridge)

Research question: Which congestion control algorithm(s) should be implemented in electrical distribution networks? Simulation model:

● arrival process (Poisson process),● congestion management protocols (proportional fairness, max-flow).

Results:● If the inter-arrival time becomes too short, the charging of vehicles takes too

long, and at some point more cars arrive for charging than leave fully charged. Hence at the critical value of the arrival rate the system undergoes a continuous phase transition from free-flow phase to congested phase.

● We study numerically the critical value of the arrival rate for realistic networks.

● We validate our findings by analysing critical arrival rate on small network for the simplified setup.

R. Carvalho, L. Buzna, R. Gibbens, F. Kelly, Critical behaviour in charging of electric vehicles, New J. Phys. (17): 095001, 2015

Simulation model: Discrete simulator

Pythoncvxopt solver

Optimization model: Modelling assumptions

● Electric vehicles arrive (Poisson process) with empty batteries,choose with uniform probability to connect to a node and leave when fully charged.

● Tree-like network topology (distribution networks) → No need to apply Kirhhoff's voltage law on network loops.

● One feeder node.● Batteries of vehicles are modelled as elastic loads (i.e. able to absorb any value of

power they are allocated by the network) and the state of the battery is time integral of allocated power.

Optimization model: Voltage drop

From here we get:

Optimization model

The first set of constraints sets the limits on the nodal voltage.

The second set of constraints is the physical law coupling the voltage to power for a subtree.

It is quadratic, hence it is not convex! So we make a substitution of variables (SD relaxation).

Max-flow: Proportional fairness:

Optimization model: Convex relaxation

We replace variables V with W variables and to make the substitution equivalent we add for each matrix the constraints that ensure that it is rank one and positive semidefinite.

Rank one constraint is not convex! However, (in this case) removing rank one constraint does not change the Pareto frontier or the optimum.

[L. Gan, N. Li, U. Topcu, S. Low, Exact Convex Relaxation of Optimal Power Flow in Radial Networks, IEEE Trans. Autom. Control, 60, 72-87, 2012]

Optimization model


Numerical experimentsFor Poisson arrival rate when < c, all vehicles that arrive with empty batteries within a large enough time window, leave fully charged within that period (free flow phase);

For > c, some vehicles have to wait for increasingly long times to fully charge (congested phase);

We characterise this behaviour by order parameter that represents the ratio at the steady state between the uncharged vehicles and the number of vehicles that arrive to be charged:

where and indicates an average over time . in the free-flow phase, up to some critical arrival rate whereas congested phase is identified by .

[A. Arenas, A. Diaz-Guilera, R. Guimera, Communication in networks with hierarchical branching, Phys. Rev. Lett. 86, 3196-3199, 2001]

SCE networks

*[L. Gan, N. Li, U. Topcu, S. Low, Exact Convex Relaxation of Optimal Power Flow in Radial Networks, IEEE Trans. Autom. Control, 60, 72-87, 2012]

Node indexes identify the edges and resistance and reactance is taken from reference*. Node 1 is the root node in both networks. Nodes 13, 17, 19, 23, 24 (in lighter colour) are photovoltaic generators and we removed them from the network.

We set Vnominal = B = 1.0 and = 0.1.

Onset of congestion

Onset of congestion

The number N(t) of charging vehicles at time t may fluctuate widely close to the critical point, and thus it is not easy to determine c. To alleviate this limitation we adopt the susceptability-like measure*:

where is the length of a time window and is the standard deviation of the order parameter . To compute , we first consider a long time series and split it into windows with length . We next determine the value of order parameter in each window, and finally calculate the standard deviation of these values.

*[A. Arenas, A. Diaz-Guilera, R. Guimera, Communication in networks with hierarchical branching, Phys. Rev. Lett. 86, 3196-3199, 2001]

Onset of congestion

Inequality in charging timeWe compute Gini coefficient of charging time.

Gini coefficient is typically used as a measure of inequality in income distribution. Gini coefficient is one half of the mean difference in units of the mean:

where u and v are independent identically distributed random variables with probability density f and mean .

For a sample (xi, i = 1, 2, … , n), the gini coefficient may be estimated by sample mean:

For sample with one non-zero value x we get:

For sample where all data points have the same value we get:

Inequality in charging time

Gini coefficient values for income:Sweden: G=0.26 U.S.: G=0.41Seychelles: G=0.66

Onset of congestion in 2-edge network We assume a time interval t.


t 1MF :

P2MF P3

MF=0

t 2MF :

P2MF=0 P3

MF

t 1PF :

P2PF P3

PF

t 2PF :

P2PF=0 P3

MF

R , X R , X

Onset of congestion in 2-edge networkMax-flow: Proportional fairness:P2

MF t1MF=

BΛc Δ t2

P3MF t 2

MF=BΛc Δ t

2

P2PF t1

PF=BΛc Δ t

2

P3MF t 2

PF=BΛc Δ t

2−P3

PF t 1PF

P2MF=

2α(1−α)V nominal2

R

P3MF=

α(1−α)V nominal2

R

P2PF=

2α V nominal2 (3√γ−γ)

9 R

P3PF=

(1−α)V nominal2 (√γ+3α−3)

3 R

γ=2√α2−α+1|α−2|+2α2−5 α+5

Simplification: Power allocations in PF are independent of car number, i.e. . w1=w2=1.0

λcMF=

P2MF

B2

(P2

MF

P3MF +1)

λcPF=

P3MF

B2

(P3

MF

P2PF −

P3PF

P2PF +1)

Parameter values : R = X = B = Vnominal = 1.0, = 0.1.

Onset of congestion in 2-edge network

Current and future workOur next goal is to study how the future electrical distribution networks could evolve to support heterogeneous types of loads (uncooperative and cooperative network users) using self-organized (market) mechanism.The network should allow interaction between heterogeneous populations of users by providing mechanisms that could be used to provide active network users with necessary information and the correct incentives to use the network in a fair and efficient way.

What should be done ?

- reconsideration of model assumption (upper bounds for power allocations neglected reactive parts of the voltage drops, neglected thermal limits of power lines, tree-like network topology);

- enhancement of the user model (utility function, user strategy);

- enhancement of the user models using relevant field data.

Workshop co-organized by the ERAdiate project

Please, feel free to ask.

Acknowledgements

This lecture was supported by the research grants VEGA 1/0463/16 "Economically efficient charging infrastructure deployment for electric vehicles in smart cities and communities", APVV-15-0179 "Reliability of emergency systems on infrastructure with uncertain functionality of critical elements" and it was facilitated by the FP 7 project ERAdiate [621386]"Enhancing Research and innovation dimensions of the University of Zilina in Intelligent Transport Systems".

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Equitable distribution of resources in Transportation Networks · Optimization model: Convex relaxation We replace variables V with W variables and to make the substitution equivalent